UNITEXT for Physics

Paul R. Berman

Introductory Quantum MechanicsA Traditional Approach Emphasizing Connections with Classical Physics

UNITEXT for Physics

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Paul R. Berman

Introductory QuantumMechanicsA Traditional Approach EmphasizingConnections with Classical Physics

123

Paul R. BermanUniversity of MichiganAnn Arbor, MI, USA

ISSN 2198-7882 ISSN 2198-7890 (electronic)UNITEXT for PhysicsISBN 978-3-319-68596-0 ISBN 978-3-319-68598-4 (eBook)https://doi.org/10.1007/978-3-319-68598-4

Library of Congress Control Number: 2017954936

© Springer International Publishing AG 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

I would like to dedicate this book toDebra Berman, my wife of 30+ years,for her support and positive outlook on life.

Preface

This book is based on junior and senior level undergraduate courses that I havegiven at both New York University and the University of Michigan. You might ask,in heavens name, why anyone would want to write yet another introductory texton quantum mechanics. And you would not be far off base with this assessment.There are many excellent introductory quantum mechanics texts. Moreover, with thematerial available on the internet, you can access almost any topic of your choosing.Therefore, I must agree that there are probably no compelling reasons to publish thistext. I have undertaken this task mainly at the urging of my students, who felt that itwould be helpful to students studying quantum mechanics.

For the most part, the approach taken is a traditional one. I have tried toemphasize the relationship of the quantum results with those of classical mechanicsand classical electromagnetism. In this manner, I hope that students will be ableto gain physical insight into the nature of the quantum results. For example, in thestudy of angular momentum, you will see that the absolute squares of the sphericalharmonics can be given a relatively simple physical interpretation. Moreover, byusing the effective potential in solving problems with spherical symmetry, I amable to provide a physical interpretation of the probability distributions associatedwith the eigenfunctions of such problems and to interpret the structures seen inscattering cross sections. I also try to stress the time-dependent aspects of problemsin quantum mechanics, rather than focus simply on the calculation of eigenvaluesand eigenfunctions.

The book is intended to be used in a year-long introductory course. Chapters 1–13 or 1–14 can serve as the basis for a one-semester course. I do not introduceDirac notation until Chap. 11. I do this so students can try to master the wavefunction approach and its implications before engaging in the more abstractDirac formalism. Dirac notation is developed in the context of a more generalapproach in which different representations, such as the position and momentumrepresentations, appear on an equal footing. Most topics are treated at a levelappropriate to an undergraduate course. Some topics, however, such as the hyperfineinteractions described in the appendix of Chap. 21, are at a more advanced level.These are included for reference purposes, since they are not typically included in

vii

viii Preface

undergraduate (or graduate) texts. There is a web site for this book, http://www-personal.umich.edu/~pberman/qmbook.html, that contains an Errata, Mathematicasubroutines, and some additional material.

The problems form an integral part of the book. Many are standard problems, butthere are a few that might be unique to this text. Quantum mechanics is a difficultsubject for beginning students. I often tell them that falling behind in a course suchas this is a disease from which it is difficult to recover. In writing this book, myforemost task has been to keep the students in mind. On the other hand, I knowthat no textbook is a substitute for a dedicated instructor who guides, excites, andmotivates students to understand the material.

I would like to thank Bill Ford, Aaron Leanhardt, Peter Milonni, Michael Revzen,Alberto Rojo, and Robin Shakeshaft for their insightful comments. I would also liketo acknowledge the many discussions I had with Duncan Steel on topics containedin this book. Finally, I am indebted to my students for their encouragement andpositive (as well as negative) feedback over the years. I am especially grateful tothe Fulbright foundation for having provided the support that allowed me to offer acourse in quantum mechanics to students at the College of Science and Technologyat the University of Rwanda. My interactions with these students will always remainan indelible chapter of my life.

Ann Arbor, MI, USA Paul R. Berman

http://www-personal.umich.edu/~pberman/qmbook.htmlhttp://www-personal.umich.edu/~pberman/qmbook.html

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Radiation Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Wave Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 BlackBody Spectrum: Origin of the Quantum Theory . . . . . . . . . . . . 71.3 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Bohr Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 De Broglie Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 The Schrödinger Equation and Probability Waves. . . . . . . . . . . . . . . . . . 151.7 Measurement and Superposition States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7.1 What Is Truly Strange About QuantumMechanics: Superposition States . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.7.2 The EPR Paradox and Bell’s Theorem . . . . . . . . . . . . . . . . . . 181.8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.9 Appendix: Blackbody Spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.9.1 Box Normalization with Field Nodes on the Walls . . . . . . 221.9.2 Periodic Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.9.3 Rayleigh-Jeans Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.9.4 Planck’s Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.9.5 Approach to Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.1 Complex Function of a Real Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Functions and Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.1 Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.2 Scalar Functions of Three Variables. . . . . . . . . . . . . . . . . . . . . . 362.2.3 Vector Functions of Three Variables . . . . . . . . . . . . . . . . . . . . . 37

2.3 Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4 Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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2.6 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Free-Particle Schrödinger Equation: Wave Packets . . . . . . . . . . . . . . . . . . . 533.1 Electromagnetic Wave Equation: Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2 Schrödinger’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2 Free-Particle Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4 Schrödinger’s Equation with Potential Energy: Introductionto Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Time-Independent Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4 Appendix: Schrödinger Equation in Three Dimensions . . . . . . . . . . . . 734.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Postulates and Basic Elements of Quantum Mechanics:Properties of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.1 Hermitian Operators: Eigenvalues and Eigenfunctions. . . . . . . . . . . . . 78

5.1.1 Eigenvalues Real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1.3 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1.4 Continuous Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.1.5 Relationship Between Operators . . . . . . . . . . . . . . . . . . . . . . . . . 885.1.6 Commutator of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.1.7 Uncertainty Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.1.8 Examples of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2 Back to the Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2.1 How to Solve the Time-Dependent Schrödinger

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.2.2 Quantum-Mechanical Probability Current Density . . . . . . 1025.2.3 Operator Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2.4 Sum of Two Independent Quantum Systems . . . . . . . . . . . . 105

5.3 Measurements in Quantum Mechanics: “Collapse”of the Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.5 Appendix: From Discrete to Continuous Eigenvalues . . . . . . . . . . . . . . 1095.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Problems in One-Dimension: General Considerations, InfiniteWell Potential, Piecewise Constant Potentials, and DeltaFunction Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.1.1 Potentials in which V.x/ > 0 and V .˙1/ � 0 . . . . . . . . . 116

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6.1.2 Potentials in which V.x/ > 0 and V .�1/ � 0while V .1/ � 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1.3 Potentials in which V.x/ > 0 and V .�1/ � 0while V .1/ D W > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.1.4 Potentials in which V.x/ > 0 and V .˙1/ � 1 . . . . . . . . 1186.1.5 Potentials in which V.x/ < 0 and V .˙1/ � 0 . . . . . . . . . 118

6.2 Infinite Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.2.1 Well Located Between �a=2 and a=2 . . . . . . . . . . . . . . . . . . . . 1226.2.2 Well Located Between 0 and a . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.2.3 Position and Momentum Distributions . . . . . . . . . . . . . . . . . . . 1256.2.4 Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3 Piecewise Constant Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.3.1 Potential Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.3.2 Square Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376.3.3 Potential Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.4 Delta Function Potential Well and Barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.4.1 Square Well with E < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.4.2 Barrier with E > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.6 Appendix: Periodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.6.1 Bloch States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.7.1 Advanced Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7 Simple Harmonic Oscillator: One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 1637.1 Classical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.2 Quantum Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

7.2.1 Eigenfunctions and Eigenenergies . . . . . . . . . . . . . . . . . . . . . . . 1687.2.2 Time-Dependent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

7.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1757.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8 Problems in Two and Three-Dimensions: General Considerations . . . 1798.1 Separable Hamiltonians in x; y; z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

8.1.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.1.2 Two- and Three-Dimensional Infinite Wells . . . . . . . . . . . . . 1808.1.3 SHO in Two and Three Dimensions. . . . . . . . . . . . . . . . . . . . . . 181

8.2 General Hamiltonians in Two and Three Dimensions . . . . . . . . . . . . . . 1828.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9 Central Forces and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.1 Classical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1859.2 Quantum Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.2.1 Angular Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1879.2.2 Physical Interpretation of the Spherical Harmonics . . . . . 196

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9.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2019.4 Appendix: Classical Angular Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 2019.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

10 Spherically Symmetric Potentials: Radial Equation. . . . . . . . . . . . . . . . . . . . 20710.1 Radial Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21010.2 General Solution of the Schrödinger Equation for

Spherically Symmetric Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21210.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

10.3 Infinite Spherical Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21510.4 Finite Spherical Well Potential: Bound States . . . . . . . . . . . . . . . . . . . . . . 22310.5 Bound State Coulomb Problem (Hydrogen Atom) . . . . . . . . . . . . . . . . 22510.6 3-D Isotropic Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23510.7 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24010.8 Appendix: Laguerre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24010.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

11 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24911.1 Vector Spaces and Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

11.1.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25011.1.2 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25311.1.3 Schrödinger’s Equation in Momentum Space . . . . . . . . . . . 261

11.2 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26311.2.1 Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

11.3 Angular Momentum Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27011.4 Solving Problems Using Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

11.4.1 Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27411.4.2 Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27411.4.3 Angular Momentum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27511.4.4 Limited Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

11.5 Connection with Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27611.5.1 Time Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

11.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.7 Appendix A: Matrix Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28311.8 Appendix B: Spherical Harmonics in Dirac Notation . . . . . . . . . . . . . . 28411.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

12 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29112.1 Classical Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29112.2 Spin Magnetic Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29212.3 Spin-Orbit Coupling in Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29812.4 Coupling of Orbital and Spin Angular Momentum. . . . . . . . . . . . . . . . . 29912.5 Spin and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30112.6 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30412.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

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13 (A) Review of Basic Concepts (B) Feynman Path IntegralApproach (C) Bell’s Inequalities Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30913.1 Review of Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

13.1.1 Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30913.1.2 Wave Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31013.1.3 Quantum-Mechanical Current Density . . . . . . . . . . . . . . . . . . . 31313.1.4 Eigenfunctions and Eigenenergies . . . . . . . . . . . . . . . . . . . . . . . 31313.1.5 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31413.1.6 Measurement in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 31513.1.7 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31513.1.8 Heisenberg Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

13.2 Feynman Path-Integral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31813.3 Bell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

13.3.1 Proof of Bell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32213.3.2 Electron Spin Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32413.3.3 Photon Polarization Measurements. . . . . . . . . . . . . . . . . . . . . . . 32613.3.4 Quantum Teleportation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32913.3.5 Why the Big Fuss? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

13.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33113.5 Appendix: Equivalence of Feynman Path Integral

and Schrödinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33213.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

14 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33914.1 Non-degenerate Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340

14.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34314.2 Degenerate Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35014.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35514.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

15 Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36115.1 Combining Perturbation Theory and the Variational Approach. . . . 36515.2 Helium Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36815.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37115.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

16 WKB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37516.1 WKB Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37516.2 Connection Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

16.2.1 Bound State Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38016.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38116.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38416.5 Appendix: Connection Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38516.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

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17 Scattering: 1-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39117.1 Steady-State Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39417.2 Time-Dependent Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

17.2.1 E0 > V0 for Step Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39717.2.2 E0 < V0 for Step Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

17.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40517.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

18 Scattering: 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40918.1 Classical Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40918.2 Quantum Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

18.2.1 Method of Partial Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42418.2.2 Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

18.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45218.4 Appendix A: Free Particle Solution in Spherical Coordinates . . . . . 45218.5 Appendix B: Hard Sphere Scattering when ka� 1 . . . . . . . . . . . . . . . . 45518.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

19 Symmetry and Transformations: Rotation Matrices . . . . . . . . . . . . . . . . . . . 46119.1 Active Versus Passive Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

19.1.1 Passive and Active Translations . . . . . . . . . . . . . . . . . . . . . . . . . . 46219.1.2 Passive and Active Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46319.1.3 Extension to Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 466

19.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46919.2.1 Translation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47019.2.2 Rotation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47219.2.3 Extension to Include Spin Angular Momentum . . . . . . . . . 47619.2.4 Transformation of Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

19.3 Rotations: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47919.3.1 Rotation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48019.3.2 Rotation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48119.3.3 Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

19.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48319.5 Appendix: Finite Translations and Rotations . . . . . . . . . . . . . . . . . . . . . . . 483

19.5.1 Translation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48319.5.2 Rotation Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

19.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

20 Addition of Angular Momenta, Clebsch-Gordan Coefficients,Vector and Tensor Operators, Wigner-Eckart Theorem . . . . . . . . . . . . . . . 49120.1 Addition of Angular Momenta and Clebsch-Gordan

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49120.2 Vector and Tensor Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49520.3 Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50120.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50320.5 Appendix: Proof of the Wigner-Eckart Theorem . . . . . . . . . . . . . . . . . . . 50320.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

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21 Hydrogen Atom with Spin in External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 51121.1 Energy Levels of Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

21.1.1 Unperturbed Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51321.1.2 Relativistic Mass Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51321.1.3 Spin Included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51621.1.4 Spin–Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51621.1.5 Darwin Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51921.1.6 Total Fine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52021.1.7 Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52121.1.8 Hyperfine Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52121.1.9 Addition of a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52321.1.10 Addition of an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

21.2 Multi-Electron Atoms in a Magnetic Field: Vector Model . . . . . . . . . 53121.2.1 Zeeman Splitting Much Greater Than Spin-Orbit

Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53121.2.2 Spin-Orbit Splitting Much Greater Than Zeeman

Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53221.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53521.4 Appendix A: Radial Integrals for Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . 53521.5 Appendix B: Hyperfine Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

21.5.1 Hyperfine Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54021.5.2 Zeeman Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54321.5.3 Contribution to the Hyperfine Splitting from

Non-contact Interaction Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 54621.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551

22 Time-Dependent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55522.1 Time-Dependent Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555

22.1.1 Interaction Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55722.2 Spin 1/2 Quantum System in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . 558

22.2.1 Analytic Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56022.3 Two-Level Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

22.3.1 Rotating-Wave or Resonance Approximation . . . . . . . . . . . 56622.3.2 Analytic Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56822.3.3 Field-Interaction Representation . . . . . . . . . . . . . . . . . . . . . . . . . 571

22.4 Density Matrix for a Single Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57322.4.1 Magnetic Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57922.4.2 Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58022.4.3 Selection Rules for Electric Dipole and Magnetic

Dipole Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58322.5 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58722.6 Appendix: Interaction of an Atom with a Thermal Bath . . . . . . . . . . . 58722.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

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23 Approximation Techniques in Time-Dependent Problems . . . . . . . . . . . . . 59323.1 Time-Dependent Perturbation Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59323.2 Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

23.2.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59823.3 Sudden Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60023.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60123.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

24 Decay of a Discrete State into a Continuum of States: Fermi’sGolden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60724.1 Discrete State Coupled to a Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

24.1.1 Photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61024.1.2 Spontaneous Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

24.2 Bounded Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62024.3 Zeno Paradox and Zeno Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62224.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62524.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

Chapter 1Introduction

As a science or engineering major, you are about to embark on what may be themost important course of your undergraduate career. Quantum mechanics is thefoundation on which our current picture of the structure of matter is built. For stu-dents, an introductory course in quantum mechanics can be difficult and frustrating.When you studied Newtonian physics, it was easy to envision experiments involvingthe motion of particles moving under the influence of forces. Electromagnetismand optics were a bit more abstract once the concept of fields was introduced,but you are familiar with many optical effects such as colors of thin films, therainbow, and diffraction from single or double slits. In quantum mechanics, it ismore difficult conceptually to understand what is going on since you have very littleday to day experience with the wave nature of matter. However, you can exploit yourknowledge of both classical mechanics and optics to get a better feel for quantummechanics.

I often begin a course in quantum mechanics asking students “Why and when isquantum mechanics needed to explain the dynamics of a particle moving in a poten-tial. In other words, when does a classical particle description fail?” To understandand appreciate many aspects of quantum mechanics, you should always have thisand a few other questions in the back of your mind. Question two might be “Howis the quantum mechanical solution of a given problem related to the correspondingproblem in classical mechanics?” Question three could be “Is there a problem inoptics with which I am familiar that can shed light on a given problem in quantummechanics?” You can add other questions to this list, but these are a good start.

Another key point in the study of quantum mechanics is not to lose track ofthe physics. There are many special functions, such as Hermite and Laguerrepolynomials, that emerge from solutions of the Schrödinger equation, and thealgebra can get a little complicated. As long as you remind yourself of the physicalnature of the solution rather than the mathematical details, you will be in great

© Springer International Publishing AG 2018P.R. Berman, Introductory Quantum Mechanics, UNITEXT for Physics,https://doi.org/10.1007/978-3-319-68598-4_1

1

https://doi.org/10.1007/978-3-319-68598-4_1

2 1 Introduction

shape. The reason for this is that most of the solutions share many common features.The specific form of the solutions may change, but the overall qualitative nature ofthe solutions is remarkably similar for many different problems. Moreover, withthe availability of symbolic manipulation programs such as Mathematica, Maple,or Matlab, it is now easy to plot and evaluate any of these functions using a fewkeystrokes.

To help introduce the subject matter, I will present a very broad, qualitativeoverview of the way in which quantum mechanics was born, a birth that took about25–30 years. I will not worry about historical accuracy here, but simply try to giveyou a reasonable picture of the manner in which it became appreciated that a wavedescription of matter was needed in certain limits.

1.1 Electromagnetic Waves

Quantum mechanics is a wave theory. Since the wave properties of matter havemany similarities to the wave properties of electromagnetic radiation, it won’t hurtto review some of the fundamental properties of the electromagnetic field. Thepossible existence of electromagnetic waves followed from Maxwell’s equations.By combining the equations of electromagnetism, Maxwell arrived at a waveequation, in which the wave propagation speed in vacuum was equal to v D1=p�0�0, where �0 is the permittivity and �0 the permeability of free space. Since

these were known quantities in the nineteenth century, it was a simple matter tocalculate v; which turned out to equal the speed of light. This result led Maxwell toconjecture that light was an electromagnetic phenomenon.

All electromagnetic waves travel in vacuum with speed c D 2:99792458 �108 m/s, now defined as the speed of light. What distinguishes one type ofelectromagnetic wave from another is its wavelength � or frequency f , which arerelated by

c D f�: (1.1)

Instead of characterizing a wave by its frequency (which has units of cycles persecond or Hz) and wavelength, we can equally well specify the angular frequency,! D 2� f , (which has units of radians per second or s�1) and the magnitude of thepropagation vector (or wave vector), defined by k D 2�=�. With these definitions,Eq. (1.1) can be replaced by

! D ck; (1.2)

which is known as a dispersion relation, relating frequency to wave vector. Forelectromagnetic radiation in vacuum the dispersion relation is linear.

1.1 Electromagnetic Waves 3

The source of electromagnetic waves is oscillating or accelerating charges, whichgive rise to propagating electric and magnetic fields. The wave equation for theelectric field vector E.R; t/ at position R at time t in vacuum is

r2E.R; t/ D 1c2@2E.R; t/@t2

: (1.3)

The simplest solution of this equation is also the most important, since it is abuilding block solution from which all other solutions can be constructed. Thebuilding block solution of the wave equation is the infinite, monochromatic, planewave solution, having an electric field vector given by

E.R; t/ D �E0 cos .k � R � !t/ ; (1.4)

where E0 is the field amplitude and the polarization of the field is specified by aunit vector � that is perpendicular to the propagation vector k of the field. There aretwo independent field polarizations possible for each propagation vector. The factthat ��k D 0 follows from the requirement that r � E.R;t/ D 0 in vacuum. Thefield (1.4) corresponds to a wave that is infinite in extent and propagates in the kdirection. Of course, no such wave exists in nature since it would uniformly fill allspace.

As a consequence of the linearity of the wave equation, the sum of any twosolutions of the wave equation is also a solution. This is known as a superpositionprinciple. It is not difficult to visualize how waves add together. If, at the same timeyou are putting your finger in and out of water in a lake, someone else is also puttingtheir finger in and out of the water, the resulting wave results from the actions ofboth your fingers. The important thing to remember is that it is the displacementsor amplitudes of the waves that add, not their intensities. The intensity of a wave isproportional to the square of the wave amplitude.

For the time being, let us consider two waves having the same frequency and thesame amplitude. If the two waves propagate in opposite directions, the total electricfield vector is

E.R; t/ D �E0 Œcos .k � R � !t/C cos .�k � R � !t/�D 2�E0 cos .!t/ cos .k � R/ ; (1.5)

a standing wave pattern. If you plot the wave amplitude at several different timesas a function of position, you will find that there is an envelope for the wave thatis fixed in space—the wave “stands” there and oscillates within the envelope. Thepoints of zero amplitude are called nodes of the field and the maxima (or minima)are called antinodes of the field. For a standing wave field in the x direction that isconfined between two parallel mirrors separated by a distance L; the standing wavepattern will “fit in” for wavelengths equal to 2L; L, 2L=3; L=2, etc., as shown in

4 1 Introduction

Fig. 1.1 Resonance involving standing waves with clamped endpoints

Fig. 1.1. In this limit, the tangential component of the electric field vanishes at themirrors, as required by the boundary conditions on the field. The condition

�n D 2L=n; n D 1; 2; 3; : : : : (1.6)

is known as a resonance condition.

1.1.1 Radiation Pulses

To get a pulse of radiation, it is necessary to add together monochromatic waveshaving a continuous distribution of frequencies. It is easy to create a pulse ofradiation. Simply turn a laser or other light source on and off and you have createda pulse. Why worry about the frequencies contained in the pulse? It turns out that itis important, even central, to understand this concept if you are going to have someidea of what quantum mechanics is about.

Let us assume the pulse is propagating in the x direction and has a duration �tcorresponding to a spatial width�x D c�t. Clearly this cannot be a monochromaticwave since a monochromatic wave is not localized. Instead, the pulse must be asuperposition of waves having a range of frequencies �f centered around someaverage frequency f0. Using the theory of Fourier analysis it is possible to show that�f and �t are related by

�f�t � 12�: (1.7)

The quantity �f is known as the spectral width of the pulse. If the frequencyof the pulse is known precisely, as in a monochromatic wave, there is no rangeof frequencies and �f D 0: In general there is a central frequency f0 in a pulseand a range of frequencies �f about that central frequency as shown in Fig. 1.2.If �f � f0, then the frequency is pretty well-defined and the field is said to bequasi-monochromatic. The field associated with the light of a green laser pointer

1.1 Electromagnetic Waves 5

Fig. 1.2 Illustration of the equation �f�t � 1=2�: As the spectral frequency distribution of apulse broadens, the temporal pulse width narrows

and most other laser fields are quasi-monochromatic, as are the fields from neondischarge tubes. Radio waves are also quasi-monochromatic, the central frequencyis the frequency of the station (about 1000 kHz for AM and 100 MHz for FM)broadcasting the signal. On the other hand, if �f is comparable with f0, as in a lightbulb, the source is said to be incoherent or broadband.

For example, consider a pulse of green laser light that has an ns (10�9 s) duration,�t D 1 ns. The spatial extent of this pulse is 3 � 108 m/s � 10�9 s = 30 cm. Thefrequency range in the pulse is given by

�f D 1=�2�10�9 s

�D 1:6 � 108 Hz. (1.8)

Since the light from the green laser pointer has a central wavelength of 532 nm, ithas a central frequency of about f0 D 5:6 � 1014 Hz. As a consequence, �f � f0and this pulse is quasi-monochromatic. That is, a one nanosecond pulse of this lightappears to have a single frequency or color if you look at it. On the other hand, ifyou try to make a one femtosecond (fs) pulse with this light (1 fs = 10�15 s) whichhas a spatial extent of 300 nm, then

�f D 1=�2�10�15 s

�D 1:6 � 1014 Hz, (1.9)

which is comparable to the central frequency. As a result this is a broadband pulsewhich no longer appears green, but closer to white. Note that the spatial extent of the

6 1 Introduction

pulse, 300 nm is comparable to the central wavelength. This is an equivalent test forbroadband radiation. If the spatial extent is comparable to the central wavelength,the radiation is broadband; if it is much larger than the central wavelength, it isquasi-monochromatic.

1.1.2 Wave Diffraction

The final topic that we will need to know something about is wave diffraction.Diffraction is a purely wave phenomenon, but sometimes waves don’t appear todiffract at all. If you shine a laser beam at a tree or a pencil, it will reflect off ofthese and not bend around them. Moreover, if you shine a laser beam through anopen door, it will not bend around into the hallway. Diffraction is only importantwhen waves meet obstacles (including apertures or openings) that are comparableto or smaller than the central wavelength of the waves. This is actually true for thelaser beam itself!—it will stay a beam of approximately constant diameter only if itsdiameter is much greater than its wavelength, otherwise, it will spread significantly.

I can make this somewhat quantitative. Imagine there is a circular opening ina screen having diameter a through which a laser beam passes, or that it passesthrough a slit having width a, or that a laser beam having diameter a propagates invacuum with no apertures present. In each of these cases, there is diffraction thatleads to a spreading of the beam by an amount

sin � � �a; (1.10)

where � is the angle with which the beam spreads. If �=a � 1, the spreadingangle is very small and diffraction is relatively unimportant. This is the limit wherethe wave acts as a particle, moving on straight lines. On the other hand, when agets comparable to a wavelength, the spreading of the beam becomes significant. If�=a > 1, the spreading is over all angles. For example, if a pinhole is illuminatedwith light, you can see the diffracted light at any angle on the other side of thepinhole.

Even if the diffraction angle is small, the effects can get large over long distances.If a laser beam having diameter 1.0 cm and central wavelength 600 nm is sent to themoon, the diffraction angle is

� � �aD 6 � 10

�7 m10�2 m

D 6 � 10�5: (1.11)

By the time it gets to the moon the spot size diameter d is

d � REM� D 3:8 � 108 m � 6 � 10�5 D 2:3 � 104 m D 23 km, (1.12)

a lot larger than 1 cm! In this equation REM is the Earth-moon distance.

1.2 BlackBody Spectrum: Origin of the Quantum Theory 7

The bottom line on diffraction. If you try to confine a wave to a distance less thanor comparable to its wavelength, it will diffract significantly.

1.2 BlackBody Spectrum: Origin of the Quantum Theory

Thermodynamics was developed in the nineteenth century and involves the studyof the properties of vapors, liquids, and solids in terms of such parametersas temperature, pressure, volume, density, conductivity, etc. At the end of thenineteenth century, the theory of statistical mechanics was formulated in which theproperties of systems of particles are explained in terms of their statistical properties.It was shown that the two theories of thermodynamics and statistical mechanicswere consistent—one could explain macroscopic properties of vapors, liquids, andsolids by considering them to be made up of a large number of particles followingNewton’s laws. One result of this theory is that for a system of particles (or waves) inthermodynamic equilibrium at a temperature T, each particle or wave has .1=2/kBTof energy for each “degree of freedom.” This is known as the equipartition theorem.

The quantity kB D 1:38 � 10�23 J/ıK is Boltzmann’s constant and T is theabsolute temperature in degrees Kelvin. A “degree of freedom” is related to anindependent motion a particle can have (translation, rotation, vibration, etc.). Fora free particle (no forces acting on it), there are three degrees of freedom, onefor each independent direction of motion. For a transverse wave, there are twodegrees of freedom, corresponding to the two possible independent directions forthe polarization of the wave.

At the beginning of the twentieth century there was a problem in trying toformulate the theory of a blackbody. A blackbody is an object that, in equilibrium,absorbs and emits radiation at the same rate. At a given temperature, the radiationemitted by a blackbody is spread over a wide range of frequencies, but the peakintensity occurs at a wavelength �max governed by Wien’s law,

�maxT D 2:90 � 10�3 m �ı K, (1.13)

where T is the temperature in degrees Kelvin. For example, the surface of the sun isabout 5000 ıK. If the sun is approximated as a blackbody, then �max � 580 nm is inthe yellow part of the spectrum. As an object is heated it emits radiation at higherfrequencies; an object that is “blue” hot is hotter than an object that is “red” hot.

The experimental curve for emission from a blackbody as a function of wave-length is shown in Fig. 1.3. To try to explain this result, Rayleigh and Jeansconsidered a model for a blackbody consisting of a cavity with a small hole init. Standing wave patterns of radiation fill the cavity and each standing wave haskBT of energy associated with it (kBT=2 for each independent polarization of thefield). A standing wave pattern or mode is characterized by the number of (half)wavelengths in the x, y, and z directions. There is a maximum wavelength � D 2L ineach direction, but there is no limit on the minimum wavelength. Since the number

8 1 Introduction

Fig. 1.3 Intensity per unit wavelength (in arbitrary units) as a function of wavelength [in units of(hc=kBT)] for a blackbody. The maximum occurs at �max � hc= .5kBT/

of small wavelength (high frequency) modes that can fit is arbitrarily large and sinceeach mode has kBT of energy, the amount of energy needed to reach equilibriumbecomes infinite. Another way of saying this is that the energy density (energy perunit frequency interval) approaches infinity as � � 0 or f � 1. The fact that it doesnot take an infinite amount of energy to bring a blackbody into thermal equilibriumor that the energy density does not approach infinity at high frequencies is known asthe “ultraviolet catastrophe”; theory and experiment were in disagreement for highor “ultraviolet” frequencies (smaller wavelengths) (see Fig. 1.3).

Planck tried to overcome this difference by making an ad hoc hypothesis.1 Ineffect, he said, that to excite a mode of frequency f , a specific minimum amount ofenergy hf (h is defined below) is needed that must be provided in an all or nothingfashion by electrons oscillating in the cavity walls. This went against classical ideasthat energy can be fed continuously to build up oscillation in a given mode. Byadopting this theory, he found that he could explain the experimental data if hechose

h D 6:63 � 10�34 J � s D 4:14 � 10�15 eV � s, (1.14)

which is now known as Planck’s constant [recall that one eV (electron-volt) is equalto 1:60�10�19 J (Joules)]. The quantum theory was born. Some details of the Plancksolution are given in the Appendix.

In thermal equilibrium at temperature T , electrons in the walls of the cavity havea maximum energy of order of a few kBT , so that they are capable of exciting only

1On the law of the distribution of energy in the normal distribution, Annalen der Physik 4, 553–563(1901).

1.3 Photoelectric Effect 9

those radiation modes having hf . kBT . Quantitatively, it can be shown that themaximum in the blackbody spectrum as a function of frequency occurs at

fmax D2:82kBT

hD 5:88 � 1010T Hz/ıK. (1.15)

As a function of wavelength the maximum in the blackbody spectrum occurs at

�max Dhc

4:965kBTD 2:90 � 10

�3

Tm �ı K. (1.16)

The numerical factors appearing in these equations and the fact that

fmax�max ¤ c (1.17)

follow from the details of the Planck distribution law (see Appendix). Visibleradiation having �max D 500 nm corresponds to a blackbody temperature of T �6; 000 ıK, about the temperature of the surface of the sun. In the visible part of thespectrum, hf is of order of a few eV.

In 1907, Einstein also used the Planck hypothesis to explain a feature that hadbeen observed in the specific heat of solids as a function of temperature.2 Manysolids have a specific heat at constant volume equal to 3R at room temperature;where R D 8:31 J/mole/ıK is the gas constant; however, some substances such asdiamond have a much lower specific heat. Based on a model of solids composedof harmonic oscillators, Einstein used Planck’s radiation law to show that thecontributions to the specific heat of diamond from the oscillations are “frozen out”at room temperature. To fit the data on diamond, Einstein used a value of 2:73�1013Hz for the frequency of oscillations, which would imply that the vibrational degreesof freedom begin to diminish for temperatures below T D hf=kB � 1300 ıK andare effectively frozen out for temperatures below T D hf= .2:82kB/ � 460 ıK.

1.3 Photoelectric Effect

The all or nothing idea surfaced again in Einstein’s explanation of the photoelectriceffect in Section 8 of his 1905 paper, On a heuristic point of view concerning theproduction and transformation of light.3 It had been noticed by Hertz and J. J.Thomson in 1897 and 1899, respectively, that electrons could be ejected from metalsurfaces when light was shined on the surfaces. Einstein gave his explanation of theeffect in 1905 (this explanation and not relativity theory was noted in his Nobel Prize

2Planck theory of radiation and the theory of specific heats, Annalen der Physik 22, 180–190(1907).3Annalen der Physik 17, 132–148 (1905).

10 1 Introduction

citation) and further experiments by Millikan confirmed Einstein’s explanation.It turns out that Einstein’s explanation is not really evidence for photons, eventhough the authors of many textbooks assert this to be the case. To understandthe photoelectric effect, it is necessary to know that different metals have differentwork functions. The work function (typically on the order of several eV) is theminimum energy needed to extract an electron from a metal surface. In some senseit is a measure of the height of a hill that the electron must climb to get out of themetal. From a physical perspective, when an electron leaves the surface, it createsan image charge inside the surface that attracts the charge back to the surface; thework function is the energy needed to escape from this attractive force.

The experimental observations for the photoelectric effect can be summarized asfollows:

1. When yellow light is shined on a specific metal, no electrons are ejected.Increasing the intensity of the light still does not lead to electrons being ejected.

2. When ultraviolet light is shined on the same metal, electrons are ejected. Whenthe intensity of the ultraviolet radiation is increased, the number of electronsejected increases, but the maximum kinetic energy of the emitted electrons doesnot change.

To explain these phenomena, Einstein stated that radiation consists of particles(subsequently given the name photons by the chemist Gilbert Lewis in 1926). Forradiation of frequency f , the photons have energy hf . When such particles areincident on a metal surface, they excite the electrons by giving up their energy in anall or nothing fashion. Thus, if the work function is denoted by W, the frequency ofthe radiation must satisfy hf > W for the radiation to cause electrons to be ejected.If hf < W, the photons cannot excite the electrons, no matter what the intensityof the field—this explains why increasing the intensity of the yellow light does notlead to electrons being ejected. On the other hand, if hf > W, the photon can excitethe electron with any extra energy going into kinetic energy KE of the electrons(some of which may be lost on collisions). Thus the maximum energy of the emittedelectrons is

KEmax D hf �W: (1.18)

Increasing the intensity of the radiation does not change this maximum kineticenergy since it results from single photon events, it affects only the intensity ofthe emitted electrons.

1.4 Bohr Theory

By 1910, it was known that there existed negatively charged particles having charge�e equal to �1:6� 10�19 C and mass m equal to 9:1� 10�31 kg. Moreover, the sizeof these particles could be estimated using theoretical arguments related to their

1.4 Bohr Theory 11

energy content to be about 10�15 m. From atomic densities and measurements ingases, atoms were known to have a size of about 10�10 m. Since matter is neutral,there must be positively charged particles in atoms that cancel the negative charge.

But how were the charges arranged? Was the positive charge spread out over alarge sphere and the negative charge embedded in it or was there something likea planetary model for atoms? This question was put to rest in 1909 by Geigerand Madsen who collided alpha particles (helium nuclei that are produced inradioactivity) on thin metal foils. They found “back-scattering” that indicated thepositive charges were small. In 1911 Rutherford analyzed the data and estimatedthe positive charges to have a size on the order of 10�15 m. He then proposed aplanetary model of the atom in which the electron orbited the positive nucleus.

However, there were problems in planetary—model—ville. It is easy to cal-culate that an accelerating electron in orbit around the nucleus should radiate itsenergy in about a nanosecond, yet atoms were stable. Moreover, the spectrum ofradiation emitted by the hydrogen atom consisted of a number of discrete (quasi-monochromatic) lines, but accelerating electrons in different orbits should producea broadband source of radiation. How could this be explained?

To explain the experimental data within the context of the planetary model, Bohrin 1913 came up with the following postulates:

1. The electron orbits the nucleus in circular orbits (back to Ptolemy and Coperni-cus) having discrete values of angular momentum given by

L D mvr D n„; (1.19)

where n D 1; 2; 3; 4; : : : (actually he formulated the law in terms of energyrather than angular momentum, but the two methods yield the same results).The quantity r is the radius of the orbit, v is the electron’s speed, and L is themagnitude of the angular momentum. The mass in this equation should actuallybe the reduced mass of the electron which is smaller than the electron mass by afactor of 1=1:00054:

2. The electrons radiate energy only when they “jump” from a larger to a smallerorbit. The frequency of the radiation emitted is given by

fn2;n1 DEn2 � En1

h; (1.20)

where Eni is the energy associated with an orbit having angular momentum L Dni„.Although these are seemingly benign postulates, they have extraordinary con-

sequences. To begin with, I combine the first postulate with the laws of electricalattraction and Newton’s second law. The magnitude of the electrostatic force on theelectron produced by the proton is

F D 14��0

e2

r2: (1.21)

12 1 Introduction

Combing this with Newton’s second law, I find

1

4��0

e2

r2D ma D mv

2

r; (1.22)

or

mvr D e2

4��0: (1.23)

Equations (1.19) and (1.23) can be solved for the allowed radii and speeds of theelectron. The possible speeds are

vn D˛FSc

n; (1.24)

where

˛FS D1

4��0

e2

„c �1

137(1.25)

is known as the fine-structure constant. The allowed radii are

rn D n2a0; (1.26)

where

a0 D4��0„2

me2DN�c˛FS� 0:0529 nm (1.27)

is the Bohr radius, N�c D �c=2� , and

�c D h=mc D 2:43 � 10�12 m (1.28)

is the Compton wavelength of the electron. The allowed energies are given by

En Dmv2n2� 14��0

e2

rnD �ER

n2; (1.29)

where the Rydberg energy ER is defined as

ER D1

2mc2˛2FS � 13:6 eV D 2:18 � 10�18 J. (1.30)

Equations (1.19), (1.24), (1.26), and (1.29) are amazing in that they predictthat only orbits having quantized values of angular momentum, speed, radius, and

1.5 De Broglie Waves 13

energy are permitted, totally in contradiction to any classical dynamic models.Moreover the smallest allowed radius r1 D a0 gives the correct order of magnitudefor the size of atoms. In addition, the energies could be used to explain the discretenature of the spectrum of the hydrogen atom.

Bohr’s second postulate addresses precisely this point. Since the energies arequantized, the frequencies emitted by an excited hydrogen atom are also quantized,having possible values

fnq DEn � Eq

hD ER

h

�1

q2� 1

n2

�D 3:2880 � 1015

�1

q2� 1

n2

�Hz, (1.31)

with n > q. The corresponding wavelengths are

�nq D 91:18n2q2

n2 � q2 nm. (1.32)

The different frequencies of the radiation emitted on transitions ending on the n D 1level are referred to as the Lyman series, those ending on n D 2 as the Balmerseries, and those ending on n D 3 as the Paschen series. All Lyman transitions arein the ultraviolet or soft X-ray part of the spectrum. The three lowest frequencytransitions in the Balmer series are in the visible (red—�32 D 656 nm, blue-green—�42 D 486 nm, and violet—�52 D 434 nm), with the remaining Balmer transitionsin the ultraviolet. All Paschen transitions and those terminating on levels havingn > 3 are in the infrared or lower frequency spectral range. The second postulatealso “explained” the stability of the hydrogen atom, since an electron in the n D 1orbital had nowhere to go.

Thus, Bohr’s theory explained in quantitative terms the spectrum of hydrogen.That is, the Bohr theory gave very good agreement with the experimental valuesfor the frequencies of the emitted lines. Bohr theory leads to correct predictions forthe emitted frequencies because Bohr got the energies right. The Bohr theory alsopredicts the correct characteristic values for the radius and speed of the electron inthe various orbits. As we shall see, however, the radius and speed are not quantizedin a correct theory of the hydrogen atom. Moreover, while angular momentum isquantized in the quantum theory, the allowed quantized values for the magnitude ofthe angular momentum differ somewhat from those given by the Bohr theory.

1.5 De Broglie Waves

It turns out that the photoelectric effect and quantization conditions of the Bohrtheory of the hydrogen atom can be explained if one allows for the possibility thatmatter is described by a wave equation. The first suggestion of this type originateswith the work of Louis de Broglie in 1923. The basic idea is to explain the stable

14 1 Introduction

orbits of the electron in the hydrogen atom as a repeating wave pattern. Just whattype of wave remains to be seen. De Broglie actually arrived at a wavelength formatter using two different approaches.

First let us consider the circular orbits of the electron in hydrogen, for whichLn D mvnrn D n„: If we imagine that an orbit fits in n wavelengths of matter, then

n .�dB/n D 2�rn Dnh

mvn; (1.33)

where �dB is the de Broglie wavelength. Solving for �dB, I find

.�dB/n Dh

mvn: (1.34)

Thus, if we assign a wavelength to matter equal to Planck’s constant divided bythe momentum of the particle, then the wave will “repeat” in its circular orbit. IfPlanck’s constant were equal to zero, there would be no wave-like properties tomatter.

De Broglie reached this result in another manner starting with Einstein’s theoryof relativity, combined with aspects of the photoelectric effect. In special relativitythe equation for the length of the momentum-energy 4-vector is

p2c2 � E2 D m2c4; (1.35)

where p is the momentum, m the mass, and E the energy of the particle. If I applythis equation to “particles” of light having zero mass, I find p D E=c, which relatesthe momentum and energy of light. Considering light of frequency f to be composedof photons having energy E D hf , we are led to the conclusion that the momentumof a photon is p D E=c D hf=c D h=�, where � is the wavelength of the light. Thus,for photons, � D h=p. De Broglie then suggested that this is the correct prescriptionfor assigning a wavelength to matter as well.

You can calculate the de Broglie wavelength of any macroscopic object andwill find that it is extremely small compared with atomic dimensions. Since thede Broglie wavelength of macroscopic matter is much smaller than the size of anatom, macroscopic matter could exhibit its wave-like properties only if it encountersinteraction potentials that vary on such a scale, an unlikely scenario. On the otherhand, if you calculate the de Broglie wavelength of the electron in the ground stateof hydrogen, you will find a value close to the Bohr radius. Since the electron isconfined to a distance on the order of its wavelength, it exhibits wave-like properties.

In 1927, Davisson and Germer observed the diffraction and interference ofelectron waves interacting with a single crystal of nickel, providing confirmationof the de Broglie hypothesis. About the same time Schrödinger was formulating awave theory of quantum mechanics. The wave theory of matter was born!

1.6 The Schrödinger Equation and Probability Waves 15

1.6 The Schrödinger Equation and Probability Waves

With de Broglie’s hypothesis, it is not surprising that scientists tried to developtheories in which matter was described by a wave equation. In 1927, Schrödingerdeveloped such a theory at the same time that Heisenberg was developing a theorybased on matrices. Eventually the two theories were shown to be equivalent. I willfocus on the Schrödinger approach in this discussion; the matrix approach is similarto Dirac’s formalism, which is discussed in Chap. 11.

The Schrödinger equation is a partial differential equation for a function .r; t/which is called the wave function. Before thinking about how to interpret thewave function, it is useful to describe how to solve the Schrödinger equation. TheSchrödinger equation is different than the wave equation for electromagnetic waves,although there are some similarities. To solve the Schrödinger equation, one mustfind the “building block” solutions, that is, those solutions analogous to the infiniteplane monochromatic waves that served as the building block solutions of the waveequation in electromagnetism. The building block solutions of the Schrödingerequation are called eigenfunctions. Eigenfunctions or eigenstates are solutions ofthe Schrödinger equation for which j .r; t/j2 is constant in time. It is not obviousthat such solutions exist, but it can be shown that they always can be found. Theeigenfunctions are labeled by eigenvalues which correspond to dynamic constantsof the motion such as energy, momentum, angular momentum, etc. Let me give yousome examples.

A free particle is a particle on which no forces act. For such a particle bothmomentum p D mv and energy E D mv2=2 are constant. Thus momentum andenergy can be used as eigenvalue labels for a free particle. But which is a moreencompassing label? I can write the energy of the free particle as E D p2=2m. If Igive you the momentum, you can tell me the energy. On the other hand, if I give youthe energy you can tell me the magnitude of the momentum, but not its direction.Thus momentum uniquely determines the eigenfunction since, to each momentum,there is associated exactly one eigenstate. On the other hand, for a given energy thereis an infinite number of eigenstates since the momentum can be in any direction. Inthis case the eigenfunctions or eigenstates are said to be infinitely degenerate. Wewill see that whenever there is a degeneracy of this type, there is an underlyingsymmetry of nature. In this case there is translational symmetry, since the particlecan move in any direction without experiencing a change the force acting on it (sincethere is no force). The eigenfunctions for the free particle are our old friends, infiniteplane waves, but they differ from electromagnetic waves. The wavelength of the freeparticle waves is just the de Broglie wavelength � D h=p—one momentum impliesone wavelength. Thus it is the momentum that determines the wavelength of the freeparticle eigenfunctions.

For the bound states of the hydrogen atom, the electron is subjected to a forceso that momentum is not constant and cannot be used as a label for the hydrogenatom eigenfunctions. On the other hand, angular momentum is conserved for anycentral force. For reasons to be discussed in Chap. 9, the three components of theangular momentum vector cannot be used to label the eigenfunctions; instead, themagnitude of the angular momentum and one of its components are used.

16 1 Introduction

Let me return to the free particle. What is the significance of the wave functionj .r; t/j2? The interpretation given is that j .r; t/j2 is the probability density(probability per unit volume) to find the particle at position r at time t. To giveyou some idea of what this means, imagine throwing darts at a wall. You throw onedart and it hits somewhere on the wall. Now you throw a second and a third dart.Now you throw a million darts. After a million darts, you will have a pretty goodidea where the next dart will land—in other words, you will know the probabilitydistribution for where the darts will land. This will not tell you where the next dartwill hit, only the probability. The probability of hitting a specific point is essentiallyequal to zero—you must talk about the probability density (in this case, probabilityper unit area). In quantum mechanics we have something similar. You prepare aquantum system in some initial state. At some later time you measure the particlesomewhere. Now you start with an identical initial state, wait the same amount oftime and measure the particle again. You repeat this a very large number of times andyou will have the probability distribution to find the particle at a time t after it wasprepared in this state. This will not tell you where the particle will be the next timeyou do the experiment—only the probability. If you send single particles, acting aswaves, through the two-slit apparatus, each particles will set off only one detectoron the screen. If you repeat this many times however, eventually you will build upthe same type of interference pattern that occurs in the double slit experiment forlight.4

Since the eigenfunctions for a free particle are infinite mono-energetic (singlemomentum or velocity) waves, the probability density associated with the freeparticle eigenfunction is constant over the entire universe! In other words, theeigenfunctions of the free particle are spread out over all space. Clearly no suchstate exists. Any free particles that we observe in nature are in a superposition statethat is referred to as a wave packet, which is the matter analogue of a radiationpulse. As with radiation, we make up a superposition state by adding severalwaves or eigenfunctions together. But radiation pulses and wave packets differ ina fundamental way. All frequency components of a radiation pulse propagate atthe same speed in vacuum, the speed of light. For a free particle wave packet,the different eigenfunctions that compose the wave packet correspond to differentwavelengths, which, in turn, implies that they correspond to different momenta,since wavelength for matter waves is related to momentum. Thus different parts of afree particle wave packet move at different rates—the wave packet’s shape changesin time! In the case of matter waves, Eq. (1.7) is replaced by

�x�p „=2; (1.36)

4A beautiful video of an experiment demonstrating the buildup of a two-slit interference patternfor electrons can be found at http://www.hitachi.com/rd/portal/highlight/quantum/.

http://www.hitachi.com/rd/portal/highlight/quantum/

1.7 Measurement and Superposition States 17

where �x is the spread of positions and �p the spread of momenta in the wavepacket. Equation (1.36) is a mathematical expression of the famous HeisenbergUncertainty Principle.

Matter can be considered to be “particle-like” as long as it is not confined it toa distance less than its de Broglie wavelength. If a particle is confined to a distanceon the order of its de Broglie wavelength or encounters changes in potential energythat vary significantly in distances of order of its de Broglie wavelength, the particleexhibits wave-like properties.

1.7 Measurement and Superposition States

Measurement is one of the most difficult and frustrating features of quantummechanics. The problem is that the measuring apparatus itself must be classical andnot described by quantum mechanics. You will hear about “wave-function collapse”and the like, but quantum mechanics does not describe the dynamics in which aquantum state is measured by a classical apparatus. As long as you ask questionsrelated to the probability of one or a succession of measurements, you need not runinto any problems.

1.7.1 What Is Truly Strange About Quantum Mechanics:Superposition States

When I discussed electromagnetic waves, I showed that a superposition of planewave states can result in a radiation pulse. There is nothing unusual about a radiationpulse. However, for quantum-mechanical waves, the superposition of eigenfunctionscan sometimes lead to what at first appears to be rather strange results. Thestrangeness is most readily apparent if we look at the electrons in atoms or otherquantum particles that are in a superposition of such bound states. Consider theelectron in the hydrogen atom. It is possible for the electron to be in a superpositionof two or more of its energy states (just as a free particle can be in a superposition ofmomentum eigenstates). Imagine that you prepare the electron in a superposition ofits n D 1 and n D 2 energy states. What does this mean? If you measure the energyof the electron you will get either �13:6 eV or �3:4 eV. If you prepare the electronin the same manner and again measure the energy, you will get either �13:6 eVor �3:4 eV. After many measurements on identically prepared electrons, you willknow the relative probability that the electron will be measured in the n D 1 stateor the n D 2 state. You might think that sometimes the electron was prepared in then D 1 state and sometimes in the n D 2 state, but this is not the case since it isassumed that the electron is always prepared in an identical manner which can yieldeither measurement—that is, it is in a superposition state of the n D 1 and n D 2states. When you measure the energy, some people (myself excluded) like to say that

18 1 Introduction

the wave function collapses into the state corresponding to the energy you measure.I would say that quantum mechanics just allows you to predict the probabilities forvarious measurements and cannot provide answers to questions regarding how thewave function evolves when a measurement is made.

The idea that the observer forces a quantum system into a given state has led tosome interesting, but what I consider misleading, ideas. Perhaps the most famous isthe question of “Who killed Schrödinger’s cat?” In the “cat” scenario, a cat is putinto a box with a radioactive nucleus. If the nucleus decays, it emits a particle thatactivates a mechanism to release a poisonous gas. Since the nucleus is said to be ina superposition state of having decayed and not having decayed, the cat is also saidto be in a superposition state of being dead and alive! Only by looking into the boxdoes the observer know if the nucleus has decayed. In other words, the observationforces the nucleus into either its decayed or undecayed state. Thus it appears thatlooking into the box can result in the death of the cat.

My feeling is that this is all a lot of nonsense. The reason that it is nonsenseis somewhat technical, however. When the nucleus decays, it emits a gamma ray,so the appropriate superposition state must include this radiated field as well. Thetransition from initial state to the decayed state plus gamma ray essentially occursinstantaneously, even though the time at which the decay occurs follows statisticallaws. Thus when you look into the box, the cat is either alive or it is dead, but not ina superposition state. In other words, the health of the cat is a measure of whetheror not the nucleus has decayed, but the cat itself is not in a superposition state. Onmany similarly prepared systems, you will simply find that the cat dies at differenttimes according to some statistical law. You needn’t worry about being convicted of“catacide” if you open the box.

Note that if it were true that you could force the nucleus into a given state byobserving it, you could prevent the nucleus from decaying simply by looking at itcontinuously! Since it starts in its initial state and you continuously force it to stay inits initial state by looking at it, it never decays. This is known as the Zeno effect, inanalogue with Zeno’s paradoxes. It should be pointed out that it is possible to keepcertain quantum systems in a given state by continuous observation. As long as youmake measurements on the system on a time scale that is short compared with thetime scale with which the system will evolve into a superposition state, you can keepthe system in its initial state. For the decay of particles, however, the transition frominitial to final states occurs essentially instantaneously and it is impossible to usecontinuous “measurements” to keep the particle in its undecayed state. I return to adiscussion of the Zeno effect in Chap. 24.

1.7.2 The EPR Paradox and Bell’s Theorem

Although a single quantum system in a superposition state has no classical analogueand already represents a strange animal, things get really strange when we considertwo, interacting quantum systems that are prepared in a specific manner.

1.7 Measurement and Superposition States 19

1.7.2.1 The EPR Paradox

In 1935, Einstein, Podolsky, and Rosen (EPR) published a paper entitled CanQuantum-Mechanical Description of Physical Reality Be Considered Complete,which has played an important role in the development of quantum mechanics.5 Foran excellent account of this paper and others related to it, see Quantum Paradoxesand Physical Reality by Franco Selleri.6 In their paper, EPR first pose questions asto what constitutes a satisfactory theory: “Is the theory correct? Is the descriptiongiven by the theory complete? They then go on to define what they mean by anelement of physical reality: If, without in any way disturbing a system, we canpredict with certainty (i.e. with probability equal to unity) the value of a physicalquantity, then there exists an element of physical reality corresponding to thisphysical quantity.” Based on this definition and giving an example in which the wavefunction corresponds to an eigenfunction of one of two non-commuting operators,7

they conclude that “either (1) the quantum-mechanical description of reality givenby the wave function is not complete or (2) when the operators corresponding to twophysical quantities do not commute the two quantities cannot have simultaneousreality.”

To illustrate the EPR paradox, one can consider the decay of a particle into twoidentical particles. The initial particle has no intrinsic spin angular momentum,whereas each of the emitted particles has a spin angular momentum of 1=2 (spinangular momentum is discussed in Chap. 12). The spin of the emitted particlescan be either “up” or “down” relative to some quantization axis, but the sum ofthe components of spin relative to this axis must equal zero to conserve angularmomentum ( the spin of the original particle is zero so the total spin of the compositeparticles must be zero). The particles are emitted in opposite directions to conservelinear momentum and in a superposition of two states, a state in which one particle(particle A) has spin up and the other (particle B/ has spin down plus a state in whichone particle (particle A) has spin down and the other (particle B/ has spin up, writtensymbolically as

j i D 1p2.j"#i � j#"i/ ; (1.37)

where " refers to spin up and # to spin down. The spin states of the two particlesare said to be correlated or entangled.

Consider the situation in which the particles are emitted in this correlated stateand fly off so they are light years apart. According to quantum mechanics, the spinof each of the particles (by “spin,” I now mean either up or down) is not fixed until

5A. Einstein, B. Podolsky, and N. Rosen, Physical Review 47, 777–780 (1935).6Franco Selleri, Quantum Paradoxes and Physical R